Delay-Differential Equations. Tools for Epidemics Modelling
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Transcript of Delay-Differential Equations. Tools for Epidemics Modelling
Delay-Differential Equations. Tools for
Epidemics Modelling
𝑏𝑦 𝐼𝑔𝑛𝑎𝑠𝑖𝐺𝑟𝑜𝑠
Contents
Delay-Differential Equations
• Differential Equations• Example• Stability• Initial Conditions
• Delay-Differential Equations• Solution• Behaviour of Solutions• Stability
Model Types with Example
• SIR Model• Incubation Period Considered
• With Delay-Differential Equations• By Compartments (SEIR Model)
• Symptoms Considered (SEAIR Model)• Space Spread
Conclusions
Delay-Differential Equations
• Differential Equations• Example• Stability• Initial Conditions
• Delay-Differential Equations• Solution• Behaviour of Solutions• Stability
Differential-Delay Equations
The PendulumDifferential Equations
https://en.wikipedia.org/wiki/Rheonomous
Differential-Delay Equations
The PendulumDifferential Equations (Stability)
https://en.wikipedia.org/wiki/Rheonomous
Equilibrium Points:• º• º
Differential-Delay Equations
The PendulumDifferential Equations (Stability)
https://en.wikipedia.org/wiki/Rheonomous
Equilibrium Points:• º Stable• º Unstable
Differential-Delay Equations
A Differential Equation:Differential-Delay Equations (Initial Conditions)
Differential-Delay Equations
A Differential Equation:With one initial value, it’s enough!
Differential-Delay Equations (Initial Conditions)
Differential-Delay Equations
A Differential Equation:With one initial value, it’s enough!At time ,
Differential-Delay Equations (Initial Conditions)
Differential-Delay Equations
A Differential Equation:With one initial value, it’s enough!At time , So And the function can be “constructed”
Differential-Delay Equations (Initial Conditions)
Differential-Delay Equations
A Differential-Delay Equation:Differential-Delay Equations
Differential-Delay Equations
A Differential-Delay Equation:If at , ,
Differential-Delay Equations (Initial Conditions)
Differential-Delay Equations
A Differential-Delay Equation:If at , , then
Differential-Delay Equations (Initial Conditions)
Differential-Delay Equations
A Differential-Delay Equation:If at , , thenSo we cannot “construct” our function. And if we specify
Differential-Delay Equations (Initial Conditions)
Differential-Delay Equations
A Differential-Delay Equation:If at , , thenSo we cannot “construct” our function. And if we specify Still not enough, because we can “construct” just a little and then we don’t know .
Differential-Delay Equations (Initial Conditions)
Differential-Delay Equations
A Differential-Delay Equation:If at , , thenSo we cannot “construct” our function. And if we specify Still not enough, because we can “construct” just a little and then we don’t know .We need all values of for between -1 and 0. (Infinite Values)
Differential-Delay Equations (Initial Conditions)
Differential-Delay Equations
A Differential-Delay Equation:To solve it, we try ,
Differential-Delay Equations (Solution)
Differential-Delay Equations
A Differential-Delay Equation:To solve it, we try , and we find
Differential-Delay Equations (Solution)
Differential-Delay Equations
A Differential-Delay Equation:To solve it, we try , and we findHence
Differential-Delay Equations (Solution)
Differential-Delay Equations
A Differential-Delay Equation:To solve it, we try , and we findHenceWith infinite complex solutions.
Differential-Delay Equations (Solution)
Differential-Delay Equations
A Differential-Delay Equation:To solve it, we try , and we findHenceWith infinite complex solutions. A solution will be a linear combination, for example:
Differential-Delay Equations (Solution)
Differential-Delay Equations
A Differential-Delay Equation:To solve it, we try , and we findHenceWith infinite complex solutions. A solution will be a linear combination, for example:Or
Differential-Delay Equations (Solution)
Differential-Delay Equations
A Differential-Delay Equation and its Characteristic Equation:Let’s consider one posible solution:
Differential-Delay Equations (Behaviour of Solutions)
Differential-Delay Equations
A Differential-Delay Equation and its Characteristic Equation:Let’s consider an example:And let
.
Differential-Delay Equations (Behaviour of Solutions)
Differential-Delay Equations
A Differential-Delay Equation and its Characteristic Equation:Let’s consider an example:And let
.If for example and ,
Differential-Delay Equations (Behaviour of Solutions)
Differential-Delay Equations
A Differential-Delay Equation and its Characteristic Equation:Let’s consider an example:And let
.If for example and ,
Differential-Delay Equations (Behaviour of Solutions)
Differential-Delay Equations
A Differential-Delay Equation and its Characteristic Equation:Let’s consider an example:And let
.If for example and ,
Differential-Delay Equations (Behaviour of Solutions)
Differential-Delay Equations
A Differential-Delay Equation and its Characteristic Equation:Let’s consider an example:And let
.If for example and ,
Differential-Delay Equations (Behaviour of Solutions)
Differential-Delay Equations
A Differential-Delay Equation and its Characteristic Equation:Let’s consider an example:And let
.If for example and ,
Differential-Delay Equations (Behaviour of Solutions)
Differential-Delay Equations
A Differential-Delay Equation and its Characteristic Equation:Let’s consider an example:And let
.If for example and ,Whereas if and ,
Differential-Delay Equations (Behaviour of Solutions)
Differential-Delay Equations
A Differential-Delay Equation and its Characteristic Equation:Let’s consider an example:And let
.If for example and ,Whereas if and ,
Differential-Delay Equations (Behaviour of Solutions)
Differential-Delay Equations
A Differential-Delay Equation and its Characteristic Equation:Let’s consider an example:And let
.If for example and ,Whereas if and ,
Differential-Delay Equations (Behaviour of Solutions)
Differential-Delay Equations
A Differential-Delay Equation:Let’s consider an example:And let
.If for example and ,Whereas if and ,
Differential-Delay Equations (Stability)
Model Types with Example
• SIR Model• Incubation Period Considered
• With Delay-Differential Equations• By Compartments (SEIR Model)
• Symptoms Considered (SEAIR Model)• Space Spread
Model Types
In this model population is divided in 3 classes:SIR Model
Model Types
In this model population is divided in 3 classes: Susceptibles: People who are not infected but susceptible of being infected. Infecteds: People infected by the disease who can transmit it. Recovereds: People once infected, but now recovered and not susceptible of becoming infected again.
SIR Model
Model Types
In this model population is divided in 3 classes: Susceptibles: People who are not infected but susceptible of being infected. Infecteds: People infected by the disease who can transmit it. Recovereds: People once infected, but now recovered and not susceptible of becoming infected again.The Model is described by the following system:
SIR Model
Model Types
In this model population is divided in 3 classes: Susceptibles: People who are not infected but susceptible of being infected. Infecteds: People infected by the disease who can transmit it. Recovereds: People once infected, but now recovered and not susceptible of becoming infected again.The Model is described by the following system:for some , which are called the contagion and recovery parameters, respectively.
SIR Model
Susceptibles Infecteds
Recovereds
Model Types
This time we consider that the infected individuals don’t become infectious until a time of seconds passes. SIR Model with Incubation Period
Model Types
This time we consider that the infected individuals don’t become infectious until a time of seconds passes. Therefore the infectious individuals are those infecteds seconds before, minus the ones that have recovered in this period. To substract these, we add a factor of .
SIR Model with Incubation Period
Model Types
This time we consider that the infected individuals don’t become infectious until a time of seconds passes. Therefore the infectious individuals are those infecteds seconds before, minus the ones that have recovered in this period. To substract these, we add a factor of .At the same time, we consider that in the incubation period they cannot recover.
SIR Model with Incubation Period
Model Types
This time we consider that the infected individuals don’t become infectious until a time of seconds passes. Therefore the infectious individuals are those infecteds seconds before, minus the ones that have recovered in this period. To substract these, we add a factor of .At the same time, we consider that in the incubation period they cannot recover.The SIR Model was:for some .
SIR Model with Incubation Period
Model Types
This time we consider that the infected individuals don’t become infectious until a time of seconds passes. Therefore the infectious individuals are those infecteds seconds before, minus the ones that have recovered in this period. To substract these, we add a factor of .At the same time, we consider that in the incubation period they cannot recover.SIR when adding infectiousness delayed:for some .
SIR Model with Incubation Period
Model Types
This time we consider that the infected individuals don’t become infectious until a time of seconds passes. Therefore the infectious individuals are those infecteds seconds before, minus the ones that have recovered in this period. To substract these, we add a factor of .At the same time, we consider that in the incubation period they cannot recover.SIR when adding recovery delayed:for some .
SIR Model with Incubation Period
Model Types
This time we consider that the infected individuals don’t become infectious until a time of seconds passes. Therefore the infectious individuals are those infecteds seconds before, minus the ones that have recovered in this period. To substract these, we add a factor of .At the same time, we consider that in the incubation period they cannot recover.SIR Model with Incubation Period Considered:for some .
SIR Model with Incubation Period
Model Types
We consider again incubation period but by just introducing a new class: The Exposed. SEIR Model
Model Types
We consider again incubation period but by just introducing a new class: The Exposed. They are infected but not infectious, and they canot recover.
SEIR Model
Model Types
We consider again incubation period but by just introducing a new class: The Exposed. They are infected but not infectious, and they canot recover.SIR Model was:for some
SEIR Model
Susceptibles Infecteds
Recovereds
Model Types
We consider again incubation period but by just introducing a new class: The Exposed. They are infected but not infectious, and they canot recover.The Model consists on the following system:for some
SEIR Model
Susceptibles
Exposed
Infected
Recovered
Model Types
Now, following our last model, for some
SEAIR Model
Susceptibles
Exposed
Infected
Recovered
Susceptibles
Exposed
Asymptomatic
Infected
Recovered
Model Types
Now, following our last model, we introduce the class of the Asymptomatics.for some
SEAIR Model
Susceptibles
Exposed
Asymptomatic
Infected
Recovered
Model Types
Following our last model, we introduce the class of the Asymptomatics. They are infectious but with a different rate of contagion because they don’t have symptoms.The Model consists on the following system:for some
SEAIR Model
Susceptibles
Exposed
Asymptomatic
Infected
Recovered
Model Types
Following our last model, we introduce the class of the Asymptomatics. They are infectious but with a different rate of contagion because they don’t have symptoms. And they recover at a different rate, as their infection is unknown.The Model consists on the following system:for some
SEAIR Model
Susceptibles
Exposed
Asymptomatic
Infected
Recovered
Model Types
Following our last model, we introduce the class of the Asymptomatics. They are infectious but with a different rate of contagion because they don’t have symptoms. And they recover at a different rate, as their infection is unknown. There’s probability of getting symptoms.The Model consists on the following system:for some
SEAIR Model
Susceptibles
Exposed
Asymptomatic
Infected
Recovered
Model Types
Following our last model, we introduce the class of the Asymptomatics. They are infectious but with a different rate of contagion because they don’t have symptoms. And they recover at a different rate, as their infection is unknown. There’s probability of getting symptoms. And we define as the rate at which asymptomatic develop symptoms.The Model consists on the following system:for some
SEAIR Model
Model Types
Our last model is the simplest model that considers space as a factor.Space Spread
Model Types
Our last model is the simplest model that considers space as a factor. We have 2 places.Space Spread
Model Types
Our last model is the simplest model that considers space as a factor. We have 2 places. And a disease (Let’s say Ebola).Space Spread
Model Types
Our last model is the simplest model that considers space as a factor. We have 2 places. And a disease (Let’s say Ebola).Space Spread
Barcelona Sierra Leone
Model Types
Our last model is the simplest model that considers space as a factor. We have 2 places. And a disease (Let’s say Ebola).We have travel between both places and disease is spread from place 1 to place 2, and viceversa.
Space Spread
Barcelona Sierra Leone
Model Types
Our last model is the simplest model that considers space as a factor. We have 2 places. And a disease (Let’s say Ebola).We have travel between both places and disease is spread from place 1 to place 2, and viceversa. Disease is spread inside the planes (or the cars, or the trains, etc.) as well.
Space Spread
Barcelona Sierra Leone
Model Types
Our last model is the simplest model that considers space as a factor. We have 2 places. And a disease (Let’s say Ebola).We have travel between both places and disease is spread from place 1 to place 2, and vice-versa. Disease is spread inside the planes (or the cars, or the trains, etc.) as well. We have an SEAIR system in every region and in every plane, with interaction between them.
Space Spread
Barcelona Sierra Leone
(𝑆𝐸𝐴𝐼𝑅)𝐵 (𝑆𝐸𝐴𝐼𝑅)𝑆𝐿
(𝑠𝑒𝑎𝑖𝑟 )𝑆𝐿→𝐵
(𝑠𝑒𝑎𝑖𝑟 )𝐵→𝑆𝐿